Area of a Triangle: Formula, Examples, Definition

Area of Triangle: In a two-dimensional plane, the area of a triangle is the region enclosed by it. A triangle, as we all know, is a closed-form with three sides and three vertices. As a result, the area of a triangle equals the total space occupied by the triangle’s three sides. Half of the product of the triangle’s base and height is the usual formula for calculating its area.

The region occupied inside the boundary of a flat object or figure is defined as “area” in general. The measurement is done in square units, with square metres being the usual unit (m2). There are pre-defined formulas for calculating area for squares, rectangles, circles, triangles, and other shapes. We will study the area of triangle formulae for several sorts of triangles, as well as some example problems, in this post.

What is a Triangle?

A triangle is a polygon that has three sides. It can also be defined by a figure bounded or enclosed by three-line segments. Clearly, a triangle will have three sides and three vertices.

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Classifications of Triangles

Triangles are classified in two main ways:
a) classification based on the length of sides of the triangle
b) classification based on the internal angles of the triangle.

Classification of Triangles Based on the Length of Sides

Based on the length of the sides, triangles are classified into three types: scalene triangle, isosceles triangle, and equilateral triangle.

Scalene Triangles

If all the three sides of the triangle are different in length or if none of the sides of the triangle is equal, then the triangle is called a scalene triangle. The triangle given below is a scalene triangle. In this triangle, all three angles have different measures.

Isosceles Triangles

If any two of the three sides of a triangle are equal, then the triangle is called an isosceles triangle.

In the above triangle, the two sides that are equal are indicated. It is an isosceles triangle. In an isosceles triangle, the two angles opposite to the two equal sides are equal in measure.

Equilateral Triangle

If all the three sides of a triangle have the same length, then the triangle is called an equilateral triangle.

In an equilateral triangle, the measure of all the angles is also same. It is \({60^{\rm{o}}}.\)

Classifications of Triangles Based on Angles

Based on the measure of the internal angles, triangles are classified into three types: acute-angled triangle, right-angled triangle, an obtuse-angled triangle.

Acute-angled Triangle

If all the three angles in a triangle are acute angles, then the triangle is called an acute-angled triangle.
(An angle measuring more than \({0^{\rm{o}}}\) but, less than \({90^{\rm{o}}}\) is called acute angle)

Right-angled Triangle

If one angle in a triangle is a right-angle, then the triangle is called a right-angled triangle.
(An angle whose measure is exactly \({90^{\rm{o}}}\) is called a right-angled triangle.)

Obtuse-angled Triangle

If one angle in a triangle is an obtuse-angle, then the triangle is called an obtuse-angled triangle.
(An angle whose measure is more than \({90^{\rm{o}}}\) and less than \({180^{\rm{o}}}\) is called an obtuse-angled triangle.)

Terms Related to Area of a Triangle

Altitude of a Triangle

The perpendicular drawn from any vertex to the side opposite to the vertex is called the altitude of the triangle from that vertex.

In the above figure, a perpendicular \(AD\) is drawn from the vertex \(A\) on the side \(BC.\) So, \(AD\) is called the altitude of the triangle.

In the above figure, perpendiculars \(AD,\,BE\) and \(CF\) are drawn from the vertices \(A,\,B\) and \(C\) on the opposite sides \(BC,\,CA\) and \(AB,\) respectively. In this case, \(AD\) is considered altitude of triangle from vertex \(A\) with respect to base \(BC.\) Similarly, \(BE\) and \(CF\) are considered altitudes of triangle from vertex \(B\) and \(C\) with respect to bases \(CA\) and \(AB,\) respectively. Three altitudes of a triangle are always concurrent. The common point is called Orthocenter of the triangle.

Sometimes the perpendicular drawn from a vertex do not land on the opposite side. It lies on the extended opposite side.
In the above figure, the perpendicular \(AD\) is drawn on the extended \(BC.\) In this case \(AD\) is considered altitude of the triangle \(ABC\) with respect to base \(BC.\)

Median of a Triangle

A median is defined as a line segment that joins the vertex of a triangle and the middle point of the opposite side of the triangle.

In the above figure if, \(D\) is the mid-point of \(BC,\) then \(AD\) is called the median drawn from the vertex \(A\) on the opposite side \(BC.\)

Three medians can be drawn in any triangle from each of the vertices on the opposite sides as shown in the figure above. In any triangle, the three medians meet at a point. The point of intersection of the medians in a triangle is called the centroid of the triangle.

What is the Area of a Triangle?

The area of a triangle is the region or space enclosed by the three sides of the triangle.

Formula of Area of Triangle

There are several formulas used to calculate the area of a triangle. These are discussed below:

Area of a Triangle When Base and Height Are Given

Area of a triangle \({\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{ \times base \times height}}\)

The above formula is used when the length of any side and the corresponding height is known or given.
For the above figure, the area of the triangle \(= \frac{1}{2} \times {\rm{base \times height}} = \frac{1}{2} \times BC \times AD = \frac{1}{2} \times b \times h\)

Area of a Right Triangle

The above formula is used in right-angled triangles directly.

Hence, the formula used to calculate the area of a triangle is \(\frac{1}{2} \times {\rm{base \times height}} = \frac{1}{2} \times b \times h\) where \(b\) is the base and \(h\) is the height.

Type of Triangles Where This Formula Is Used

Apart from right-angled triangles, this formula can be used to calculate the area for any type of triangle if the length of the base and height are given or can be derived from the given information about the triangle.

Area of Triangle With 3 Sides: Heron’s Formula

This formula is used to calculate the area of a triangle if the length of all three sides is known or given.

According to this formula, the area of a triangle is given by,
\({\rm{area}} = \sqrt {s(s – a)(s – b)(s – c)} \) where, \(a,\,b\) and \(c\) are the lengths of sides of the triangle and \(s\) is the semi-perimeter of the triangle, given by \(s = \frac{{a + b + c}}{2}.\)

Area of an Equilateral Triangle

Heron’s formula can be used to derive a special formula applicable to calculate the area of an equilateral triangle.

In an equilateral triangle, all three sides are equal in length. So, in this case \(a = b = c.\)
So, \(s = \frac{{a + b + c}}{2} = \frac{{a + a + a}}{2} = \frac{{3\,a}}{2}\)

\( = \sqrt {\frac{{3a}}{2} \times \frac{a}{2} \times \frac{a}{2} \times \frac{a}{2}} = \sqrt {\frac{{3{a^4}}}{{16}}} = \frac{{\sqrt 3 }}{4}{a^2}\)

where (a) is the length of the side of the equilateral triangle.

Hence, the area of an equilateral triangle \( = \frac{{\sqrt 3 }}{4} \times {a^2} = \frac{{\sqrt 3 }}{4} \times {({\rm{ side }})^2}\)

Area of an Isosceles Triangle

Let the two equal sides of an isosceles triangle \(ABC\) be \(AB = AC = a\) and the length of the base be \(BC = b\)

Draw \(AD \bot BC\) So, \(D\) bisects \(AB\)

Hence, \(BD = \frac{b}{2}.\)

Applying the Pythagoras theorem on \(\Delta ABD\) it can be written that \(A{D^2} + B{D^2} = A{B^2}\)

\( \Rightarrow A{D^2} + {\left( {\frac{b}{2}} \right)^2} = {\left( a \right)^2}\)

\( \Rightarrow A{D^2} = {a^2} – \frac{{{b^2}}}{4}\)

\( \Rightarrow AD = \sqrt {{a^2} – \frac{{{b^2}}}{4}}\)

Hence, the area of the isosceles triangle \( = \frac{1}{2} \times {\rm{base \times height}} = \frac{1}{2} \times BC \times AD\)

\( = \frac{1}{2} \times b \times \sqrt {{a^2} – \frac{{{b^2}}}{4}} \)

Area of a Triangle When Coordinates of Its Vertices Are Given

The area of a triangle can be calculated if the coordinates of the three vertices of a triangle on a Cartesian plane are given.

In the triangle \(ABC\) shown above, \(A\left( {{x_1},\,{y_1}} \right),\,B\left( {{x_2},\,{y_2}} \right)\) and \(C\left( {{x_3},\,{y_3}} \right)\) are the coordinates of the vertices of the triangle.

The area of this triangle can be calculated by using the formula,
\( {{area}} = \frac{1}{2}\left| { {x_1}\left({{y_2} – {y_3}} \right) + {x_2}\left({{y_3} – {y_1}} \right) + {x_3}\left({{y_1} – {y_2}} \right)} \right|\)

Relation Between Areas of Two Triangles Lying on the Same Base and Between Same Parallels

Interesting relation exists between the areas of two different triangles if they lie on the same base and they are between the same two parallel lines. It can be proved that the area of two triangles having the same base and lying between the same two parallel lines are equal.

Here, the line \(BC\) is parallel to the line \(DQ.\) Hence, clearly, \(\Delta ABC\) and \(\Delta PBC\) have the same base \(BC\) and they are lying between the same parallel lines \(BC\) and \(DQ.\)
So, the areas of \(\Delta ABC\) and \(\Delta PBC\) are equal. Therefore, \({\mathop{\rm area}\nolimits} \Delta ABC = {\rm{area}}\Delta PBC.\)

Relation Between Area of a Triangle and the Area of a Parallelogram Lying on the Same Base and Between Same Parallels

Interesting relation also exists between the area of a triangle and the area of a parallelogram, if they lie on the same base and between the same two parallel lines. It can be proved that the area of the triangle is half the area of the parallelogram if they have the same base and lie between the same two parallel lines.

Here, the line \(AB\) is parallel to the line \(QD\) Hence, clearly, \(\Delta ABC\) and the parallelogram \(ABDC\) and \(ABPQ\) have the same base \(AB\) and they are lying between the same parallel lines \(AB\) and \(QD\)

So, the areas of \(\Delta ABC\) is half the area of parallelogram \(ABDC\) and the parallelogram \(ABPQ\) as well. Therefore, \({\rm{area}}\,\Delta ABC{\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{ \times area}}\,{\rm{of}}\,{\rm{parallelogram}}\,ABCD\) and \({\rm{area}}\Delta ABC = {\mathop{\rm area}\nolimits} \Delta PBC.\)

Units Used for the Area of a Triangle

In the CGS system, a unit of area of a triangle is \({\rm{c}}{{\rm{m}}^2}\) and in SI system the unit of area of a triangle is \({{\rm{m}}^{\rm{2}}}\)

Solved Examples

Q.1. Find the area of a right-angled triangle whose lengths of the sides other than the hypotenuse are \({\rm{4\,cm}}\) and \({\rm{7cm}}\)
Ans: If the lengths of the sides other than the hypotenuse are \({\rm,{4\,cm}}\) and \({\rm{7cm}}\) then one of length must be the height and the other length will be the base. So, the area of the triangle \( = \frac{1}{2} \times {\rm{base \times height}} = \frac{1}{2} \times 4 \times 7 = 14\;{\rm{c}}{{\rm{m}}^2}.\)

Q.2. Find the area of a triangle whose lengths of the sides are \({\rm{3\,cm}}\).\({\rm{4\,cm}}\) and \({\rm{5\,cm}}\)
Ans: Here, the lengths of the three sides are given. So, we shall use Heron’s formula to calculate the area of the triangle. Here, \(a = 3\;{\rm{cm}},b = 4\;{\rm{cm}}\) and \(c = 5\,{\rm{cm}}\)According to this formula, the area of a triangle is given by,\({\rm{area}} = \sqrt {s(s – a)(s – b)(s – c)} ,\)where, \(a,\,b\) and \(c\) are the lengths of sides of the triangle and \(s\) is the semi-perimeter of the triangle, given by \(s = \frac{{a + b + c}}{2}.\)So, \(s = \frac{{a + b + c}}{2} = \frac{{3 + 4 + 5}}{2} = \frac{{12}}{2} = 6\;{\rm{cm}}\)Hence, the area of the given triangle\( = \sqrt {s\left( {s – a} \right)\left( {\left( {s – b} \right)\left( {s – c} \right)} \right)} = \sqrt {6\left( {6 – 3} \right)\left( {6 – 4} \right)\left( {6 – 5} \right)} \)\(= \sqrt {6 \times 3 \times 2 \times 1} = \sqrt {36} = 6\;{\rm{c}}{{\rm{m}}^2}\)

Q.3. Find the area of an equilateral triangle of side \(\rm,{{8cm}}{{.}}\)
Ans: Here, \(a = 8\;{\rm{cm}}\)
Hence, the area of an equilateral triangle with side \( = \frac{{\sqrt 3 }}{4} \times {a^2} = \frac{{\sqrt 3 }}{4} \times {(8)^2} = \frac{{\sqrt 3 }}{4} \times 64 = 16\sqrt 3 \;{\rm{c}}{{\rm{m}}^2}\)

Q.4. The equal sides of an isosceles triangle are \(5\;{\rm{cm}}\) each and the base is \(2\;{\rm{cm}}\) Find the area of the triangle.
Ans: Here, \(a = 5\;{\rm{cm}},b = 2\;{\rm{cm}}\)

Hence, the area of the isosceles triangle \( = \frac{1}{2} \times b \times \sqrt {{a^2} – \frac{{{b^2}}}{4}} = \frac{1}{2} \times 2 \times \sqrt {{5^2} – \frac{{{2^2}}}{4}}\)
\( = 1 \times \sqrt {25 – \frac{4}{4}} = \sqrt {25 – 1} = \sqrt {24} = \sqrt {2 \times 2 \times 2 \times 3} = 2\sqrt 6 \;{\rm{c}}{{\rm{m}}^2}\)

Q.5. Find the area of the triangle if the base is \(5\,{{cm}}\) and height along the base is \(8\,{{cm}}{{.}}\)
Ans: Area of the triangle \(= \frac{1}{2} \times {\rm{base \times height}} = \frac{1}{2} \times 5 \times 8 = 20\;{\rm{c}}{{\rm{m}}^2}]\)

Summary

In a two-dimensional plane, the area of a triangle is defined as the total space occupied by its three sides. The area of a triangle is equal to half the product of its base and height, therefore A = 1/2 b h is the fundamental formula. This formula works for any triangle, whether it’s a scalene triangle, an isosceles triangle, or an equilateral triangle. It’s important to remember that a triangle’s base and height are perpendicular to each other.

This article helps to comprehensively learn about how to calculate the area of different kinds of triangles depending on what information is available on the triangle. Knowing this one can calculate the area of any triangular shape land or any other triangular shaped item.

It also helps in calculating the area of any regular or irregular polygonal shaped land or any other item by dividing the polygon into triangles by diagonals and then obtaining the area of each triangle and adding them.

FAQs on Area of Triangle

Q.1. How many altitudes can a triangle have?
Ans: A triangle can have \(3\) (three) altitudes.

Q.2. How to find the area of a triangle if height is not given, but the length of three sides are given?
Ans: If the height is not given, but the length of three sides is given, then one can use Heron’s formula.
According to this formula, the area of a triangle is given by,
\({\rm{area}} = \sqrt {s(s – a)(s – b)(s – c)} ,\)
where, \(a,b\) and \(c\) are the lengths of sides of the triangle and \(s\) is the semi-perimeter of the triangle, given by \(s = \frac{{a + b + c}}{2}.\)

Q.3. Which formula is used normally to find the area of a right-angled triangle, when the length and the height are given?
Ans: If the length of the base and the height are known, then we can use the formula \({\rm{area = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{ \times base \times height}}{\rm{.}}\)

Q.4. For calculation of the area of a triangle using the formula \(\frac{{\rm{1}}}{{\rm{2}}}{\rm{ \times base \times height}}\) how do we use the formula if the height or the altitude from the opposite vertex do not land on the base?
Ans: Draw the altitude on the extended base and use its length to calculate the area of a triangle.

Q.5. What is the unit of area of a triangle if the sides are given in the unit \(inch\)?
Ans: \({\rm{inc}}{{\rm{h}}^{\rm{2}}}\)

Q.6. What is the other name of Heron’s formula?
Ans: Hero’s formula

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We hope you were able to get some assistance with triangles. The various topics covered here should have given you a good understanding of the definition, formula, and types of triangles, as well as their applications. If you have any questions about triangles or in general about this page, reach us through the comment box below and we will get back to you as soon as possible.